An electro-optical system converts radiant energy of a particular wavelength or range of wavelengths into an electrical signal that may be used to construct an array of integer values that represents an image. The system usually includes an optical component that focuses radiant energy onto a sensor element. The formed image on the sensing element is converted into a measurable electrical signal. One example of an electro-optical system is a consumer grade digital camera. Another example of an electro-optical system, operating outside the visible range, is the AXAF orbiting observatory which focuses radiant energy from distant galaxies in the x-ray band onto a sensor array. Both of these systems include two basic elements of an electro-optical imaging system, namely, an electronic sensing element and a component for forming an image.
The capability of a electro-optical system to resolve fine details of an object is referred to as its resolving power or resolution, while sharpness and/or acutance is often used to describe spatial image quality. The contrast in the spatial modulation of an image falls to nearly zero, as the spatial frequency increases towards the optical cutoff frequency. This may be seen as the capability of the electro-optical system to resolve details of line pairs that are spatially separated by different, ever smaller, spatial distances. The modulus of the optical transfer function (OTF), referred to as the modulation transfer function (MTF), is a metric used to evaluate the resolution or contrast of an electro-optical system, as it approaches its resolution limit.
Methods for mathematically describing a transfer function for optical systems were developed in the 1930's and 1940's as an alternative to geometric ray tracing methods, in order to better describe the wave nature of optical systems and account for the many possible aberrations present in optical systems. Empirical methods, such as trial and error, were often used because of the complexity in making these types of measurements. In the 1950's and 1960's, methods with reduced complexity were developed and the MTF as a metric for image resolving power became more prevalent.
In the 1960's, a method using a single sharp contrast edge was developed, commonly known as the knife-edge method. The knife-edge method produced a continuous MTF curve from a single edge transition in an image, whereas other methods such as harmonic analysis and contrast analysis produced performance results at only discrete spatial frequencies on the MTF curve. The knife-edge method is still the preferred method for determining the characteristic spatial performance of an optical system.
The MTF is a useful qualitative measure of the resolution of an electro-optical system and the ability of a system to produce a sharp image at specific spatial frequencies. Unfortunately, the quantitative accuracy of the MTF is generally accepted to be quite poor, with an uncertainty of 5% or more at the sensor sampling Nyquist spatial frequency, with increasing uncertainty as the spatial frequency approaches the sampling cutoff frequency of the sensor sampling function. Many of the simplifications, applied in the 1960's and still used today, ignore key features of the transfer function, in order to avoid complexity. These key features are deemed unnecessary for an inexpensive assessment of spatial performance. Furthermore, additional error tends to be introduced and the true signal tends to be lost with this process of computing MTF. Conventional techniques apply some arbitrary smoothing of the image data in an attempt to remove noise. As a result, meaningful information may be lost and apparent information, not actually present, may be introduced.
One example of using a special image performance metric is to specify resolution as a requirement for a sensor system in terms of an expected MTF value at the sampling Nyquist frequency. This is particularly important for the remote sensing industry, where customers have requirements at specific spatial frequencies. These requirements must be communicated in a consistent manner to potential suppliers, and suppliers must have a way to demonstrate the resolution capabilities of their components. The MTF is also a useful tool in comparing various imaging systems to determine the system best suited for a particular task. In these cases, it is common to specify MTF at multiple spatial frequencies, for example, five to ten spatial frequencies that are fractions of the Nyquist spatial frequency of a sensor system.
There are two conventional methods for calculating MTF using knife-edge targets. Both methods derive the edge response data from the knife-edge target data. Once the edge response data is available, both of the conventional methods for calculating a knife-edge MTF involve smoothing the edge response data through a statistical smoothing curve fit, or a piecewise fit. Then data points are sampled from the fit, at smaller spatial intervals than available in the original edge data set.
One conventional MTF calculation method from an edge response is shown in FIG. 1. The method, designated as 10, uses a derivative and a Fast Fourier Transform (FFT). The image 11 of an edge from a knife-edge type target is used to estimate edge slope and location of the edge slope by obtaining multiple data pairs (d, E) as edge response data, as shown in step 12. Next, step 13 constructs a numerical derivative of the edge response. The numerical derivative is used as an estimate of the underlying line spread function. Step 14 smooths the numerical derivative of the edge response (d, E) to obtain a curve of f(d). Step 15 subdivides the d axis into a large number of equally spaced points along the axis. Step 16 applies an FFT to the numerical derivative estimate of the line spread function to obtain the MTF at spatial frequencies determined by the sample size and spacing. The smoothing performed on the original edge response data, however, introduces error and spurious signal content into the data. The error and the signal content is further exaggerated through the numerical differentiation process.
This conventional MTF calculation method does not provide MTF for many spatial frequencies of interest, because of the limited data sample size and spacing. The MTF at the desired spatial frequencies must be obtained through interpolation between available points. This adds another source of error beyond those already described, because the available spatial frequencies may not be particularly close together, and the other sources of error from smoothing and numerical differentiation may actually distort the shape of the MTF curve.
The second conventional method for estimating MTF also works directly with the larger resampled edge data set, using an algorithm related to the 1965 work of Berge Tatian (Tatian's method). The method, designated as 20 in FIG. 2, uses an image 11 of an edge from a knife-edge type target to estimate edge slope and location of the edge slope by obtaining multiple data pairs (d, E) as edge response data, as shown in step 12. Step 21 puts a continuous curve through the data pairs (d, E). Step 22 subdivides the d axis into a large number of equally spaced points along the axis. The edge response data is centered, smoothed in some fashion, and resampled at equally spaced intervals. In order to have the edge response data centered at a pivot point for the edge, it is frequently necessary to either throw away data from one of the ends of the edge, or to add fictitious data to achieve equal numbers of data points about this pivot point. This adjustment is performed by step 23. Tatian's method is then applied in step 24.
The preparation of the data for method 20 is as prone to error as the previously described method 10. Specifically, the phase information from the OTF is lost, and only a phase shifted estimate of the MTF with unknown phase shift remains. These methods of preparing the edge response data by smoothing and resampling, followed by tweaking of the data, and the method of obtaining a pivot point for application of Tatian's method create large errors that cannot be estimated at spatial frequencies as small as 50% of the Nyquist frequency. Even worse, the data manipulation creates a process that is not generally reproducible.
A hallmark of both conventional methods is that the results are difficult to reproduce and have poor accuracy. Another major drawback of these methods is sensitivity to the phase of the target and subsequent failure to capture the phase information available in the OTF. If multiple measurements are made with the target displaced by a fraction of a sampling interval, or by a non-integral number of sampling intervals, the data generally yield different results. This further complicates obtaining good MTF estimates and adds to the difficulty in obtaining repeatable results.
Generally, an MTF curve is accepted as a metric of spatial resolving performance having value up to the Nyquist frequency, where the error is expected to be as much as 5%. Beyond the Nyquist frequency, the error is generally accepted to be greater than 5%. Furthermore, the MTF as a metric describes the qualitative resolving power of the system and phase information is ignored. Without the phase information, the uncertainty beyond the Nyquist frequency quickly becomes unacceptable. It is this uncertainty which makes cascading MTF curves problematic either in an attempt to isolate individual MTF components, or in combining components to form a complete system model. Due to the drawbacks and limitations associated with conventional test and calculation methods, MTF is used with the understanding that the error is large and perhaps unacceptable beyond the Nyquist frequency.
The present invention addresses the above deficiencies by providing a system and method of determining edge slope and location of a knife-edge image. The present invention also provides a method for determining an OTF that preserves phase information and provides good estimates of the MTF measurement error at any spatial frequency up to the optical cutoff of the system. As will be explained, the present invention measures OTF and obtains the MTF of an electro-optic system with improved accuracy and with repeatable results.